
Journal of Convex Analysis 15 (2008), No. 4, 803818 Copyright Heldermann Verlag 2008 On Semicontinuity of ConvexValued Multifunctions and Cesari's Property (Q) Andreas Löhne Institut für Mathematik, MartinLutherUniversität, TheodorLieserStraße 5, 06099 Halle, Germany andreas.loehne@mathematik.unihalle.de [Abstractpdf] \newcommand{\R}{\mathbb{R}} We investigate two types of semicontinuity for setvalued maps, Painlev\'{e}Kuratowski semicontinuity and Cesari's property (Q). It is shown that, in the context of convexvalued maps, the concepts related to Cesari's property (Q) have better properties than the concepts in the sense of Painlev\'{e}Kuratowski. In particular we give a characterization of Cesari's property (Q) in terms of upper semicontinuity of a family of scalar functions $\sigma_{f(\,\cdot\,)}(y^*) \colon X \to \overline\R$, where $\sigma_{f(x)} \colon Y^*\to \overline\R$ is the support function of the set $f(x)$. We compare both types of semicontinuity and show their coincidence in special cases. Keywords: Semicontinuity, setvalued maps, property (Q). MSC: 47H04,58C07 [ Fulltextpdf (177 KB)] for subscribers only. 